The astonishing claims are being investigated by the Mutual UFO Network (MUFON), which is the world’s biggest organisation dedicated to UFO and alien research.
A report sent to MUFON said the incident happened at Landers, California, during a major UFO event.
The people reported seeing two light sources, that they believed could have been two objects, being tailed by the military aircraft.
The sighting happened at about 10pm local time on May 21, one day of the Contact in the Desert, three-day special event in Joshua Tree, California.
A night sky watch was carried out using night vision goggles with video camera after the conference.
The person who led the skywatch and submitted the report to US-based MUFON has not been named.
Their report said: “Luckily one of my participants caught this and many of our other sightings, from all three nights no less, with his black-and-white night vision video camera.
“His additional footage of some of the other incredible sightings will be posted at a future date once they are compiled.
“Some of my participants and I saw a large, amber-coloured light low to the horizon just behind Giant Rock from where my whole group of about 50-plus people were gathered and using 15 pairs of night vision goggles.
“A contingency of about 20 of us ran to get a better view of it. To the naked eye it appeared to be a large, orangish (amber) light, that, quoting some of the 20 people; ‘flew like a bumble bee.’”
The UFO was described as looking like “a large diamond-shaped UFO.” through night-vision goggles.
An illustration of the UFO was also submitted.
She said: “This object then flew behind a small hill right behind the entire sky watch larger group.
“As those of us who ran off walked back to join the others, three military jets flew over in the direction of the object now gone from sight behind the hill.
“Some of us commented that if they were following it, meaning the jets, they were a few minutes too late.”
The group then reportedly heard the sound of a low flying jet from behind the hill.
She added: “Suddenly clearing the top of the hill and right over our heads, came a very fast military jet chasing a faster, large, red-glowing light.
“The military jet had both red and white blinking strobe lights and glowing afterburners and was very loud as it flew directly above our heads.
“The large red light had no blinking lights and no sound at all.
“It became clear to all 50-plus witnesses, this was no jet.
“Those folks present who looked with the aid of the goggles said they were seeing either two lights, or one object with two light sources on it.”
She said a review of the footage left her convinced during the single jet incident, there were two UFOs flying in perfect tandem with each other.
She wrote: “It appears you can see stars between them in the background, suggesting that these are two separate objects.
“As is usually the case, this footage, while incredible, does not do the sighting justice. “The video is in black-and-white night vision, not full-colour, naked-eye vision, and does not have any sound, so you are unable to hear the palatable excitement of 50-plus witnesses screaming and cheering.”
California MUFON Chief Investigator Earl Grey Anderson is investigating.
Roger Marsh, MUFON director of communications urged for caution before the probe ends.
He said: “Please remember that most UFO sightings can be explained as something natural or man-made.”
Scott Brando, who runs debunking website ufoofinterest.org. was less than impressed with the footage Express.co.uk presented him with.
He felt it was just two aircraft, as the chasing object also had two lights.
He said: “I see just two different aircrafts flying with different navigation lights.
“Unfortunately footage taken in night vision doesn’t help us to identify correctly those aircrafts, but I don’t see any mystery.”